DOCSLIB.ORG

Lecture 4: Normalized Functions of Bounded Variation; Regular Borel Measures; the Riesz–Markov–Kakutani Theorem;

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 4: Normalized Functions of Bounded Variation; Regular Borel Measures; the Riesz–Markov–Kakutani Theorem; Functional analysis Lecture 4: Normalized functions of bounded variation; regular Borel measures; the Riesz{Markov{Kakutani theorem; Remark 3.10. Assume that the functional Λ above is positive (i.e. Λf ≥ 0 for f ≥ 0). We observe that the extension Λe is also positive and hence the constructed function f is then monotone increasing. Indeed, suppose that there is g 2 `1[0; 1] with g(t) ≥ 0 for each t 2 [0; 1] but Λeg < 0. By normalizing, we can assume that kgk1 = 1, thus 0 ≤ g(t) ≤ 1 for each t 2 [0; 1]. Let 1 be the constant 1 function. As Λ is positive and linear, it is monotone (i.e. Λg ≤ Λh if g ≤ h), and hence Λ1 = kΛk. We have k1 − gk1 ≤ 1 and Λ(e 1 − g) = Λ1 − Λeg = kΛk − Λeg > kΛk = kΛek; which gives a contradicition. Therefore, positive continuous functionals on C[a; b] correspond to increasing func- tions on [a; b], while general continuous functionals correspond to functions of bounded variation. Remark 3.11. Since every f 2 BV([a; b]) is the difference of two increasing functions, we see that every Λ 2 (C[a; b])∗ can be written as Λ = Λ+ − Λ−; where Λ+; Λ− are positive. (3.1) The conclusion of Remark 3.11 can be justified without appealing to Riesz' Theo- rem 3.9. To this end, let us use a more general language which will be also useful in the sequel. By an ordered vector space we mean any vector space E over R equipped with a partial order relation ≥ compatible with the algebraic operations in the sense that x ≥ y implies x + z ≥ y + z for all x; y; z 2 E and x ≥ y implies λx ≥ λy for all x; y 2 E, λ ≥ 0. We denote by E+ the positive cone of E defined by fx 2 E : x ≥ 0g. An ordered vector space which is also a lattice with respect to the given order is called a Riesz space or a vector lattice. If E is a Riesz space and _, ^ denote the lattice operations, then for any vector x 2 E we define its positive part x+, negative part x− and its modulus jxj by x+ = x _ 0; x− = (−x) _ 0; jxj = x _ (−x): We then have x = x+ − x− and jxj = x+ + x−, so this gives a decomposition of x as the difference between two positive elements. A Riesz space is called Archimedean if whenever 0 ≤ nx ≤ y for all n 2 N and some y 2 E+, then x = 0. If E is a Riesz space and a normed space with a norm k·k satisfying the condition jxj ≤ jyj =) kxk ≤ kyk for all x; y 2 E; then E is called a normed Riesz space, and in the case where the norm is complete, we call E a Banach lattice. There are plenty of examples of Banach lattices: Rn with the pointwise ordering and any of the norms k·kp (1 ≤ p ≤ 1), the space C(X) of continuous 1 functions, and Cb(X) of bounded continuous functions on any topological space X, also considered with the pointwise ordering and the supremum norm, as well as Lp(µ)-spaces under the a.e. pointwise ordering. If E is a normed Riesz space, then its dual E∗ can be ordered in a natural way by defining f ≥ g if and only if f(x) ≥ g(x) for every x 2 E+ (f; g 2 E∗). We will see (Lemma 3.18 below) that in this way we equip E∗ with a Banach lattice structure1. In particular, the decomposition (3.1) is possible in the Banach lattice (C[a; b])∗. Actually, decomposition (3.1) holds true with a further requirement that kΛk = kΛ+k + kΛ−k and in a much more general setting of the so-called order unit normed Riesz spaces (see Problem 3.22). For C[a; b] we will obtain that equality with the aid of Hahn's decom- position theorem. As we know that every continuous linear functional on the real space C[a; b] is rep- resented by a function f 2 BV[a; b], let us consider the problem of uniqueness of this representation. Obviously, f is not unique as stated in Theorem 3.9, because adding any constant function to f does not change the Riemann{Stieltjes integral. So, one normalization would be to require that e.g. f(a) = 0. By Jordan's decomposition, we have f = g −h for some increasing functions g and h, and therefore f has at most countably many points of discontinuity in (a; b). Observe that by modifying the values of f at each of these points, but keeping the variation bounded, we do not change the value of the integral in formula (3.7). Indeed, any g 2 C[a; b] is Riemann{Stieltjes integrable, hence its integral can be approximated by any sequence of integral sums over partitions with diameters converging to zero2. In particular, we can avoid each point of discontinuity of f (except the endpoints) in these partitions. Consequently, another normalization condition which does not affect the validity of (3.7) is that f(t+) = f(t) for each t 2 (a; b), i.e. f is right- continuous inside the interval (a; b) (the choice of the left or right limit is arbitrary). Every f 2 BV([a; b]) satisfying the above two normalization conditions is called a normalized function of bounded variation, and we denote the class of all such functions by • NBV([a; b]) = ff 2 BV([a; b]): f(a) = 0 and f is right-continuous in (a; b)g. By modifying the representing function f in Theorem 3.9 as described above, we can guarantee that it belongs to the class NBV([a; b]). Moreover, it turns out that in this class the function f is unique. We will obtain this as a particular case of the `uniqueness part' of Theorem 3.16 below. Before proceeding to that result, let us build a `bridge' be- tween the special Riesz Theorem 3.9 and the much more general Riesz{Markov{Kakutani Theorem 3.16. Given any function f 2 BV([a; b]), we define a measure µf on the field F consisting of finite unions of the intervals jy; x](a ≤ y < x ≤ b) which are all right-closed and left-open unless y = a in which case we take the closed interval [a; x]. Namely: • µf ([a; x]) = f(x) − f(a), • µf ((y; x]) = f(x) − f(y) for a < y < x ≤ b. 1An interested reader may consult e.g. [C.D. Aliprantis, K.C. Border, Infinite dimensional analysis. A hitchhiker's guide, 3rd edition, Springer 2006; Ch. 8 and 9]. 2See [W Rudin, Principles of mathematical analysis; Thms. 6.6 and 6.8]. 2 Notice that the measure µf is finitely additive and satisfies n n X [ µf (E) = (f(bj) − f(aj)) for E = ja1; b1] [ (aj; bj] 2 F; j=1 j=2 where a ≤ a1 < b1 ≤ a2 < b2 ≤ ::: ≤ an < bn ≤ b. Moreover, µf is bounded because f is of bounded variation. But, as we will see, µf has also another nice property of being regular, provided that f belongs to the class NBV([a; b]). Definition 3.12. Let F be a field of subsets of a topological space X and µ: F! K be a finitely additive set function, where K 2 fR; Cg. We say µ is regular if for every E 2 F and " > 0 there exist a set F 2 F with F ⊆ E and a set G 2 F with int G ⊇ E satisfying jµ(C)j < " for every C 2 F;C ⊆ G n F: If f 2 NBV([a; b]), it is possible to extend µf to a regular σ-additive measure defined on all Borel subsets of [a; b]. It follows from the following general result on extensions of measures. (Actually, we only need a simplified version of this result, as the measure µf is trivially σ-additive in the sense that there are no countably infinite, but not finite, unions of the intervals jc; d] in F.) Extension theorem. Let F be a field of subsets of some topological space X and let µ: F! K be a bounded, regular, finitely additive set function, where K 2 fR; Cg. Then, µ has a unique regular, σ-additive extension µe: σ(Σ) ! K to the σ-algebra σ(F) generated by F. For the proof see e.g. [N. Dunford, J.T. Schwartz, Linear operators (vol. 1: General theory), Wiley{Interscience 1988; Ch. III.5]. This theorem is a combination of two classical measure theory results: • the Alexandroff theorem which says that every bounded regular finitely additive (real- or complex-valued) set function on a field of subsets of a compact topological space is σ-additive; • the Carath´eodory extension theorem which guarantees that positive σ-additive mea- sures defined on a ring R can be extended to positive σ-additive measures on the σ-algebra generated by R. (It should also be noted that a measure is regular if and only the positive and negative parts of its real and imaginary parts are regular.) Lemma 3.13. If f 2 BV([a; b]), then for every t 2 [a; b) we have lim V (f; [t; t + "]) = 0: "!0+ Proof. (classes) Now, assume that f 2 NBV([a; b]). Since f is right-continuous on (a; b), we have lim"!0 f(t + j"j) = f(t) for every t 2 (a; b) and it follows that for every interval jc; d] 2 F, n n X Sn o V (f; [c; d]) = sup jµf (Ij)j: Ij 2 F;Ij \ Ik = ? for j 6= k; j=1Ij ⊆ jc; d] : j=1 3 (The right-hand side defines the variation of µf on the set jc; d]; we will introduce this notion later on.) Plainly, we can extend this formula to any set E 2 F in the sense that if E = I1 [ ::: [ Im, with Ij = jcj; dj] 2 F (1 ≤ j ≤ m), then the left-hand side is Pm j=1 V (f; [aj; bj]), whereas the right-hand side is the variation of µf on E, denoted by jµf j(E).
Load more

Recommended publications
  • Vector Measures with Variation in a Banach Function Space
    VECTOR MEASURES WITH VARIATION IN A BANACH FUNCTION SPACE O. BLASCO Departamento de An´alisis Matem´atico, Universidad de Valencia, Doctor Moliner, 50 E-46100 Burjassot (Valencia), Spain E-mail:[email protected] PABLO GREGORI Departament de Matem`atiques, Universitat Jaume I de Castell´o, Campus Riu Sec E-12071 Castell´o de la Plana (Castell´o), Spain E-mail:[email protected] Let E be a Banach function space and X be an arbitrary Banach space. Denote by E(X) the K¨othe-Bochner function space defined as the set of measurable functions f :Ω→ X such that the nonnegative functions fX :Ω→ [0, ∞) are in the lattice E. The notion of E-variation of a measure —which allows to recover the p- variation (for E = Lp), Φ-variation (for E = LΦ) and the general notion introduced by Gresky and Uhl— is introduced. The space of measures of bounded E-variation VE (X) is then studied. It is shown, amongother thingsand with some restriction ∗ ∗ of absolute continuity of the norms, that (E(X)) = VE (X ), that VE (X) can be identified with space of cone absolutely summingoperators from E into X and that E(X)=VE (X) if and only if X has the RNP property. 1. Introduction The concept of variation in the frame of vector measures has been fruitful in several areas of the functional analysis, such as the description of the duality of vector-valued function spaces such as certain K¨othe-Bochner function spaces (Gretsky and Uhl10, Dinculeanu7), the reformulation of operator ideals such as the cone absolutely summing operators (Blasco4), and the Hardy spaces of harmonic function (Blasco2,3).
    [Show full text]
  • Integral Representation of a Linear Functional on Function Spaces
    INTEGRAL REPRESENTATION OF LINEAR FUNCTIONALS ON FUNCTION SPACES MEHDI GHASEMI Abstract. Let A be a vector space of real valued functions on a non-empty set X and L : A −! R a linear functional. Given a σ-algebra A, of subsets of X, we present a necessary condition for L to be representable as an integral with respect to a measure µ on X such that elements of A are µ-measurable. This general result then is applied to the case where X carries a topological structure and A is a family of continuous functions and naturally A is the Borel structure of X. As an application, short solutions for the full and truncated K-moment problem are presented. An analogue of Riesz{Markov{Kakutani representation theorem is given where Cc(X) is replaced with whole C(X). Then we consider the case where A only consists of bounded functions and hence is equipped with sup-norm. 1. Introduction A positive linear functional on a function space A ⊆ RX is a linear map L : A −! R which assigns a non-negative real number to every function f 2 A that is globally non-negative over X. The celebrated Riesz{Markov{Kakutani repre- sentation theorem states that every positive functional on the space of continuous compactly supported functions over a locally compact Hausdorff space X, admits an integral representation with respect to a regular Borel measure on X. In symbols R L(f) = X f dµ, for all f 2 Cc(X). Riesz's original result [12] was proved in 1909, for the unit interval [0; 1].
    [Show full text]
  • (Measure Theory for Dummies) UWEE Technical Report Number UWEETR-2006-0008
    A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 UWEE Technical Report Number UWEETR-2006-0008 May 2006 Department of Electrical Engineering University of Washington Box 352500 Seattle, Washington 98195-2500 PHN: (206) 543-2150 FAX: (206) 543-3842 URL: http://www.ee.washington.edu A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 University of Washington, Dept. of EE, UWEETR-2006-0008 May 2006 Abstract This tutorial is an informal introduction to measure theory for people who are interested in reading papers that use measure theory. The tutorial assumes one has had at least a year of college-level calculus, some graduate level exposure to random processes, and familiarity with terms like “closed” and “open.” The focus is on the terms and ideas relevant to applied probability and information theory. There are no proofs and no exercises. Measure theory is a bit like grammar, many people communicate clearly without worrying about all the details, but the details do exist and for good reasons. There are a number of great texts that do measure theory justice. This is not one of them. Rather this is a hack way to get the basic ideas down so you can read through research papers and follow what’s going on. Hopefully, you’ll get curious and excited enough about the details to check out some of the references for a deeper understanding.
    [Show full text]
  • Probability Measures on Metric Spaces
    Probability measures on metric spaces Onno van Gaans These are some loose notes supporting the first sessions of the seminar Stochastic Evolution Equations organized by Dr. Jan van Neerven at the Delft University of Technology during Winter 2002/2003. They contain less information than the common textbooks on the topic of the title. Their purpose is to present a brief selection of the theory that provides a basis for later study of stochastic evolution equations in Banach spaces. The notes aim at an audience that feels more at ease in analysis than in probability theory. The main focus is on Prokhorov's theorem, which serves both as an important tool for future use and as an illustration of techniques that play a role in the theory. The field of measures on topological spaces has the luxury of several excellent textbooks. The main source that has been used to prepare these notes is the book by Parthasarathy [6]. A clear exposition is also available in one of Bour- baki's volumes [2] and in [9, Section 3.2]. The theory on the Prokhorov metric is taken from Billingsley [1]. The additional references for standard facts on general measure theory and general topology have been Halmos [4] and Kelley [5]. Contents 1 Borel sets 2 2 Borel probability measures 3 3 Weak convergence of measures 6 4 The Prokhorov metric 9 5 Prokhorov's theorem 13 6 Riesz representation theorem 18 7 Riesz representation for non-compact spaces 21 8 Integrable functions on metric spaces 24 9 More properties of the space of probability measures 26 1 The distribution of a random variable in a Banach space X will be a probability measure on X.
    [Show full text]
  • Chapter 2. Functions of Bounded Variation §1. Monotone Functions
    Chapter 2. Functions of Bounded Variation §1. Monotone Functions Let I be an interval in IR. A function f : I → IR is said to be increasing (strictly increasing) if f(x) ≤ f(y)(f(x) < f(y)) whenever x, y ∈ I and x < y. A function f : I → IR is said to be decreasing (strictly decreasing) if f(x) ≥ f(y)(f(x) > f(y)) whenever x, y ∈ I and x < y. A function f : I → IR is said to be monotone if f is either increasing or decreasing. In this section we will show that a monotone function is differentiable almost every- where. For this purpose, we first establish Vitali covering lemma. Let E be a subset of IR, and let Γ be a collection of closed intervals in IR. We say that Γ covers E in the sense of Vitali if for each δ > 0 and each x ∈ E, there exists an interval I ∈ Γ such that x ∈ I and `(I) < δ. Theorem 1.1. Let E be a subset of IR with λ∗(E) < ∞ and Γ a collection of closed intervals that cover E in the sense of Vitali. Then, for given ε > 0, there exists a finite disjoint collection {I1,...,IN } of intervals in Γ such that ∗ N λ E \ ∪n=1In < ε. Proof. Let G be an open set containing E such that λ(G) < ∞. Since Γ is a Vitali covering of E, we may assume that each I ∈ Γ is contained in G. We choose a sequence (In)n=1,2,... of disjoint intervals from Γ recursively as follows.
    [Show full text]
  • Section 3.5 of Folland’S Text, Which Covers Functions of Bounded Variation on the Real Line and Related Topics
    REAL ANALYSIS LECTURE NOTES: 3.5 FUNCTIONS OF BOUNDED VARIATION CHRISTOPHER HEIL 3.5.1 Definition and Basic Properties of Functions of Bounded Variation We will expand on the first part of Section 3.5 of Folland's text, which covers functions of bounded variation on the real line and related topics. We begin with functions defined on finite closed intervals in R (note that Folland's ap- proach and notation is slightly different, as he begins with functions defined on R and uses TF (x) instead of our V [f; a; b]). Definition 1. Let f : [a; b] ! C be given. Given any finite partition Γ = fa = x0 < · · · < xn = bg of [a; b], set n SΓ = jf(xi) − f(xi−1)j: i=1 X The variation of f over [a; b] is V [f; a; b] = sup SΓ : Γ is a partition of [a; b] : The function f has bounded variation on [a; b] if V [f; a; b] < 1. We set BV[a; b] = f : [a; b] ! C : f has bounded variation on [a; b] : Note that in this definition we are considering f to be defined at all poin ts, and not just to be an equivalence class of functions that are equal a.e. The idea of the variation of f is that is represents the total vertical distance traveled by a particle that moves along the graph of f from (a; f(a)) to (b; f(b)). Exercise 2. (a) Show that if f : [a; b] ! C, then V [f; a; b] ≥ jf(b) − f(a)j.
    [Show full text]
  • Real Analysis II, Winter 2018
    Real Analysis II, Winter 2018 From the Finnish original “Moderni reaalianalyysi”1 by Ilkka Holopainen adapted by Tuomas Hytönen February 22, 2018 1Version dated September 14, 2011 Contents 1 General theory of measure and integration 2 1.1 Measures . 2 1.11 Metric outer measures . 4 1.20 Regularity of measures, Radon measures . 7 1.31 Uniqueness of measures . 9 1.36 Extension of measures . 11 1.45 Product measure . 14 1.52 Fubini’s theorem . 16 2 Hausdorff measures 21 2.1 Basic properties of Hausdorff measures . 21 2.12 Hausdorff dimension . 24 2.17 Hausdorff measures on Rn ...................... 25 3 Compactness and convergence of Radon measures 30 3.1 Riesz representation theorem . 30 3.13 Weak convergence of measures . 35 3.17 Compactness of measures . 36 4 On the Hausdorff dimension of fractals 39 4.1 Mass distribution and Frostman’s lemma . 39 4.16 Self-similar fractals . 43 5 Differentiation of measures 52 5.1 Besicovitch and Vitali covering theorems . 52 1 Chapter 1 General theory of measure and integration 1.1 Measures Let X be a set and P(X) = fA : A ⊂ Xg its power set. Definition 1.2. A collection M ⊂ P (X) is a σ-algebra of X if 1. ? 2 M; 2. A 2 M ) Ac = X n A 2 M; S1 3. Ai 2 M, i 2 N ) i=1 Ai 2 M. Example 1.3. 1. P(X) is the largest σ-algebra of X; 2. f?;Xg is the smallest σ-algebra of X; 3. Leb(Rn) = the Lebesgue measurable subsets of Rn; 4.
    [Show full text]
  • Measure Algebras and Functions of Bounded Variation on Idempotent Semigroupso
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 163, January 1972 MEASURE ALGEBRAS AND FUNCTIONS OF BOUNDED VARIATION ON IDEMPOTENT SEMIGROUPSO BY STEPHEN E. NEWMAN Abstract. Our main result establishes an isomorphism between all functions on an idempotent semigroup S with identity, under the usual addition and multiplication, and all finitely additive measures on a certain Boolean algebra of subsets of S, under the usual addition and a convolution type multiplication. Notions of a function of bounded variation on 5 and its variation norm are defined in such a way that the above isomorphism, restricted to the functions of bounded variation, is an isometry onto the set of all bounded measures. Our notion of a function of bounded variation is equivalent to the classical notion in case S is the unit interval and the "product" of two numbers in S is their maximum. 1. Introduction. This paper is motivated by the pair of closely related Banach algebras described below. The interval [0, 1] is an idempotent semigroup when endowed with the operation of maximum multiplication (.vj = max (x, y) for all x, y in [0, 1]). We let A denote the Boolean algebra (or set algebra) consisting of all finite unions of left-open, right-closed intervals (including the single point 0) contained in the interval [0, 1]. The space M(A) of all bounded, finitely additive measures on A is a Banach space with the usual total variation norm. The nature of the set algebra A and the idempotent multiplication given the interval [0, 1] make it possible to define a convolution multiplication in M(A) which makes it a Banach algebra.
    [Show full text]
  • On the Convolution of Functions of Λp – Bounded Variation and a Locally
    IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 8, Issue 3 (Sep. - Oct. 2013), PP 70-74 www.iosrjournals.org On the Convolution of Functions of p – Bounded Variation and a Locally Compact Hausdorff Topological Group S. M. Nengem1, D. Samaila1 and M. Solomon1 1 Department of Mathematical Sciences, Adamawa State University, P. M. B. 25, Mubi, Nigeria Abstract: The smoothness-increasing operator “convolution” is well known for inheriting the best properties of each parent function. It is also well known that if f L1 and g is a Bounded Variation (BV) function, then f g inherits the properties from the parent’s spaces. This aspect of BV can be generalized in many ways and many generalizations are obtained. However, in this paper we introduce the notion of p – Bounded variation function. In relation to that we show that the convolution of two functions f g is the inverse Fourier transforms of the two functions. Moreover, we prove that if f, g BV(p)[0, 2], then f g BV(p)[0, 2], and that on any locally compact Abelian group, a version of the convolution theorem holds. Keywords: Convolution, p-Bounded variation, Fourier transform, Abelian group. I. Introduction In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions, giving the area overlap between the two functions as a function of the amount that one of the original functions is translated.
    [Show full text]
  • Fonctions on Bounded Variations in Hilbert Spaces
    Fonctions on bounded variations in Hilbert spaces Giuseppe Da Prato (SNS, Pisa) Newton Institute, March 31, 2010 Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Introduction We recall that a function u : Rn ! R is said to be of bounded variation (BV) if there exists an n-dimensional vector measure Du with finite total variation such that Z Z 1 u(x)divF(x)dx = − hF(x); Du(dx)i; 8 F 2 C0 (H; H): H H The set of all BV functions is denoted by BV (Rn). Moreover, the following result holds, see E. De Giorgi, Ann. Mar. Pura Appl. 1954. Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Theorem 1 Letu 2 L1(Rn). Then (i) , (ii). (i)u 2 BV (Rn). (ii) We have Z lim jDTt u(x)jdx < 1; (1) t!0 H whereT t is the heat semigroup Z 2 1 − jx−yj Tt u(x) := p e 4t u(y)dy: n 4πt R Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces As well known functions from BV (Rn) arise in many mathematical problems as for instance: finite perimeter sets, surface integrals, variational problems in different models in elasto plasticity, image segmentation, and so on. Recently there is also an increasing interest in studyingBV functions in general Banach or Hilbert spaces in order to extend the concepts above in an infinite dimensional situation. Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces A definition of BV function in an abstract Wiener space, using a Gaussian measure µ and the corresponding Dirichlet form, has been given by M.
    [Show full text]
  • EXISTENCE of HAAR MEASURE Most of the Measure Theory We've Done in This Class Has Been Within Subsets of Rn, Even Though Measu
    EXISTENCE OF HAAR MEASURE ARUN DEBRAY NOVEMBER 19, 2015 Abstract. In this presentation, I will prove that every compact topological group has a unique left-invariant measure with total measure 1. This presentation is for UT’s real analysis prelim class. Most of the measure theory we’ve done in this class has been within subsets of Rn, even though measures are very general tools. Today, we will talk about a measure defined on a larger class of spaces, which generalizes the usual Lesbegue measure. I will define and prove its existence; Spencer will prove its uniqueness and provide some interesting examples; and Gill will talk about an application to harmonic analysis. Definition. A topological group is a topological space G with a group structure: an identity, an associative multiplication map, and an inverse map, such that multiplication G G G and inversion G G are both continuous. × ! ! There are many examples, including Rn under addition; all Lie groups; and all finite groups (with the discrete topology). Since a topological group G has a topology, we can talk about the σ-algebra of Borel sets on it, as usual generated by the open sets. But the group structure means we can also talk about left- and right-invariance: for every g G, we have continuous maps ` : G G and r : G G, sending x gx and x xg, respectively. 2 g ! g ! 7! 7! These should be read “left (resp. right) translation by g” (or “multiplication,” or “action”). Often, we want something to be invariant under these maps. Specifically, we say that a measure is left-invariant if µ(S) = µ(g S) for all g G, and define right-invariance similarly.
    [Show full text]
  • FUNCTIONS of BOUNDED VARIATION 1. Introduction in This Paper We Discuss Functions of Bounded Variation and Three Related Topics
    FUNCTIONS OF BOUNDED VARIATION NOELLA GRADY Abstract. In this paper we explore functions of bounded variation. We discuss properties of functions of bounded variation and consider three re- lated topics. The related topics are absolute continuity, arc length, and the Riemann-Stieltjes integral. 1. Introduction In this paper we discuss functions of bounded variation and three related topics. We begin by defining the variation of a function and what it means for a function to be of bounded variation. We then develop some properties of functions of bounded variation. We consider algebraic properties as well as more abstract properties such as realizing that every function of bounded variation can be written as the difference of two increasing functions. After we have discussed some of the properties of functions of bounded variation, we consider three related topics. We begin with absolute continuity. We show that all absolutely continuous functions are of bounded variation, however, not all continuous functions of bounded variation are absolutely continuous. The Cantor Ternary function provides a counter example. The second related topic we consider is arc length. Here we show that a curve has a finite length if and only if it is of bounded variation. The third related topic that we examine is Riemann-Stieltjes integration. We examine the definition of the Riemann-Stieltjes integral and see when functions of bounded variation are Riemann-Stieltjes integrable. 2. Functions of Bounded Variation Before we can define functions of bounded variation, we must lay some ground work. We begin with a discussion of upper bounds and then define partition.
    [Show full text]